We present a quantitative network design (QND) study of the Arctic sea
ice–ocean system using a software tool that can evaluate hypothetical
observational networks in a variational data assimilation system. For a
demonstration, we evaluate two idealised flight transects derived from NASA's
Operation IceBridge airborne ice surveys in terms of their potential to
improve 10-day to 5-month sea ice forecasts. As target regions for the
forecasts we select the Chukchi Sea, an area particularly relevant for
maritime traffic and offshore resource exploration, as well as two areas
related to the Barnett ice severity index (BSI), a standard measure of
shipping conditions along the Alaskan coast that is routinely issued by ice
services. Our analysis quantifies the benefits of sampling upstream of the
target area and of reducing the sampling uncertainty. We demonstrate how
observations of sea ice and snow thickness can constrain ice and snow
variables in a target region and quantify the complementarity of combining
two flight transects. We further quantify the benefit of improved atmospheric
forecasts and a well-calibrated model.

Introduction

The Arctic climate system is undergoing a rapid
transformation. Such changes, in particular reductions in sea ice
extent, are impacting coastal communities and ecosystems and are enhancing
the potential for resource extraction and shipping. In this context, the
ability to anticipate anomalous ice conditions and in particular sea ice
hazards associated with seasonal-scale and short-term variations in ice cover
is essential. For example, in 2012, despite a long-term trend of greatly
reduced ice cover in the Chukchi Sea off Alaska's coast, ice incursions and
associated hazards led to early termination of the resource exploration
season . In this context, high-quality predictions of
the ice conditions are of paramount interest. Such predictions are typically
performed by numerical models of the sea ice–ocean system. These models are
based on fundamental equations that govern the processes controlling ice
conditions. Uncertainty in model predictions arises from four sources: first,
there is uncertainty in the atmospheric forcing data (such as wind velocity
or temperature) driving the relevant processes. Second, there is uncertainty
regarding the formulation of individual processes and their numerical
implementation (structural uncertainty). Third, there are uncertain constants
(process parameters) in the formulation of these processes (parametric
uncertainty). Fourth, there is uncertainty about the state of the system at
the beginning of the simulation (initial state).

Observational information can be exploited to reduce these
uncertainties. Currently there are
several initiatives underway to extend and consolidate the
observational network of the Arctic climate system, ranging, e.g.,
from the International Arctic Systems for Observing the Atmosphere and
Surface (IASOAS) to the Global Terrestrial Network for Permafrost
(GTN-P). Ideally, all observational data streams are interpreted
simultaneously with the process information provided by the
model to yield a consistent picture of the state of the Arctic system
that balances all the observational constraints, taking into account
the respective uncertainty ranges. Data assimilation systems that tie
into prognostic models of the Arctic system are ideal tools for this
integration task because they allow a variety of observations to be
combined with the simulated dynamics of a model.

Quantitative network design (QND) is a technique that aims to design an
observational network with optimal performance. The approach is based on work
by , who optimised the station locations for a seismographic
network. It was first applied to the climate system by , who
optimised the spatial distribution of atmospheric measurements of carbon
dioxide. A series of QND studies
demonstrated the feasibility
of the network design approach and delineated the requirements for the
implementation of the first satellite mission designed to observe atmospheric
CO2 from space the Orbiting Carbon Observatory;.
Since then, the technique has been routinely applied in the design of CO2
space missions and the
extension of the in situ sampling network for atmospheric carbon dioxide.
Recent examples focus on in situ networks over Australia
and South Africa . The design of a combined atmospheric
and terrestrial network of the European carbon cycle is addressed by
.

The present study applies the QND concept to the Arctic sea ice–ocean
system. It describes the Arctic Observational Network Design (AOND)
system, a tool that can evaluate the performance of observational
networks comprising a range of different data streams. We illustrate
the utility of the tool by evaluating the relative merits of alternate
airborne transects within the context of NASA's Operation IceBridge
, assessing
their potential to improve ice forecasts in the Chukchi Sea and along
the Alaskan coast.

Methods

Our AOND system evaluates observational networks in terms of their
impact on target quantities in a data assimilation system.
Both the data assimilation system and the
AOND system are built around the same model of the Arctic sea ice–ocean
system. Below, we first present the model, then the
assimilation system and finally the QND approach that operates on top of
this model.

NAOSIM

The model used for the present analysis is the coupled sea ice–ocean model
NAOSIM North Atlantic/Arctic Ocean Sea Ice Model;.
NAOSIM is based on version 2 of the Modular Ocean Model (MOM-2) of the
Geophysical Fluid Dynamics Laboratory. The version of NAOSIM used here has
a horizontal grid spacing of 0.5∘ on a rotated spherical grid. The
rotation maps the 30∘ W meridian onto the Equator and the North
Pole onto 0∘ E. Hence, the model's x and y directions are
different from the zonal and meridional directions, and the grid is almost
equidistant. In the vertical it resolves 20 levels, their spacing increasing
with depth from 20 to 480 m. At the southern boundary (near 50∘ N) an
open boundary condition has been implemented following ,
allowing the outflow of tracers and the radiation of waves. The other
boundaries are treated as closed walls. At the open boundary the barotropic
transport is prescribed from a coarser-resolution version of the model that
covers the whole Atlantic northward of 20∘ S .

A dynamic-thermodynamic sea ice model with a viscous-plastic rheology
is coupled to the ocean model. The prognostic variables of
the sea ice model are ice thickness, snow depth, and ice concentration. Ice
drift is calculated diagnostically from the momentum balance. Snow depth and
ice thickness are mean quantities over a grid box. The thermodynamic
evolution of the ice is described by an energy balance of the ocean mixed
layer following . Freezing and melting are
calculated by solving the energy budget equation for a single ice layer with
a snow layer. When atmospheric temperatures are below the freezing point,
precipitation is added to the snow mass; otherwise it is added to the ocean. The snow
layer is advected jointly with the ice layer. The surface heat flux is
calculated through a standard bulk formula approach using prescribed
atmospheric data and sea surface temperature predicted by the ocean model.
Owing to its low heat conductivity, the snow layer has a high impact on the
simulated energy balance . The sea ice model is
formulated on the ocean model grid and uses the same time step. The models
are coupled following the procedure devised by .

Control variables. Column 1 lists the quantities in the
control vector; column 2
gives the abbreviation for each quantity; column 3 indicates whether the
quantity is an atmospheric boundary (forcing, i.e. f) field, an initial
field (i), or a process parameter (p); column 4 gives the name of each
quantity; column 5 indicates (the standard deviation of) the prior uncertainty and the
corresponding units and provides the magnitude of the parameter
value in parenthesis, where applicable; and column 6 identifies the
position of the quantity in the control vector – for initial and
boundary values (which are differentiated by region) this position refers to the first region, while the
following components of the control vector then cover regions 2 to 9.

Atmospheric forcing (10 m wind velocity, 2 m air temperature,
2 m dew point temperature, total precipitation, and total cloud
cover) is taken from the National Centers for Environmental
Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis
. This study is based on a model integration from
1 April to 31 August 2007. The initial state of this integration is the final
state of a hindcast from January 1948 to the end of March 2007, forced by
NCEP/NCAR reanalyses and in turn initialised from Polar Science Center
Hydrographic Climatology (PHC) data (ocean temperature and
salinity), zero ocean velocities and zero snow depth, a constant ice
thickness of 2 m with 100 % ice cover where the air
temperature is below the freezing temperature of the ocean's top layer and
zero ice drift. The model's process formulations depend on a number of
uncertain parameters. Table summarises atmospheric
forcing fields and initial fields, and lists a subset of the model's relevant
process parameters.

Assimilation

The variational assimilation system NAOSIMDAS
operates through minimisation of a cost function that quantifies the fit to
all observations plus the deviation from prior knowledge on a vector of
control variables x:
J(x)=12(M(x)-d)TC(d)-1(M(x)-d)+(x-x0)TC(x0)-1(x-x0),
where M denotes the model, considered as a mapping from the control vector
to observations; d the observations with data uncertainty
covariance matrix C(d); x0 the vector of prior values of
the control variables with uncertainty covariance matrix C(x0);
and the superscript T is the transpose operator. The control variables are
typically a combination of the initial state, the atmospheric forcing and the
process parameters. The data uncertainty C(d) reflects the
combined effect of observational C(d obs) and model
error C(d mod):
C(d)2=C(d obs)2+C(d mod)2.C(d mod) captures all uncertainty
in the simulation of the observations except for the uncertainty in the
control vector because this fraction of the uncertainty is explicitly
addressed by the assimilation procedure through correction of the
control vector.

The control vector x̃ that minimises Eq. (1) achieves a balance
between the observational constraints and the prior information. The minimum
is determined through variation of the control vector (hence variational
assimilation) comprising initial and boundary conditions and process
parameters. In contrast to sequential assimilation approaches, which result
in a sequence of corrections of the state predicted by the model, the
variational approach guarantees full consistency with the dynamics imposed by
the model, as it provides an entire trajectory through the state space of the
model in response to the change in the control vector. In the case of our
model this means that we infer a trajectory that assures conservation of
mass, energy and momentum (except at the lateral domain boundaries). We note
that, in this QND study, no minimisation of Eq. (1) is required.

QND

We provide a brief description of the methodological background for
QND, which follows .
The approach is based on propagation of uncertainty from the data
to a target quantity of interest.
The target quantity may be any aspect (e.g. a prognostic or diagnostic
variable or a process parameter) that can be extracted from
a simulation with the underlying model,
for example, the sea ice concentration integrated over
a particular domain and time period.

QND proceeds in two steps. In the first step, the second derivative (Hessian)
of the cost function (Eq. ) is used to approximate the inverse
of the covariance matrix C(x) of posterior uncertainty of the
control vector, which quantifies the uncertainty ranges of the control
variables that are consistent with uncertainties in the observations and the
model. Denoting the linearisation of the model by M′, we can
approximate this posterior uncertainty by
C(x)-1=M′TC(d)-1M′+C(x0)-1.
The first term on the right-hand side quantifies the observational impact
which yields an uncertainty reduction with respect to the prior uncertainty (inverse of the second term).

When the prior uncertainty is already small, the second term is large, and a
large observational impact is required to achieve a substantial uncertainty
reduction. The observational impact is large when the observations are
highly sensitive to changes in the control variables and when the data
uncertainty is small. The first condition describes the relevance of the
observation, and the second condition its quality.

In the second step, the linearisation N′ (Jacobian) of the model N used
as a mapping from the control vector to target quantities is employed to
propagate the uncertainties in the control vector forward to
the uncertainty in a target quantity σ(y):
σ(y)2=N′C(x)N′T+σ(ymod)2.
If the model were perfect, σ(ymod) would be zero. In
contrast, if the control variables were perfectly known, the first term on
the right-hand side would be zero. Equation (4) relates the uncertainty in
control space to uncertainty in a target quantity.

Evaluating Eq. (4) for the prior uncertainty C(x0) instead of the posterior
uncertainty C(x), i.e. for a case without observational constraint,
yields a prior uncertainty for the target quantity:
σ(y0)2=N′C(x0)N′T+σ(ymod)2.
We define the term uncertainty reduction relative to σ(y0), i.e. by
σ(y0)-σ(y)σ(y0)=1-σ(y)σ(y0).
For example, if σ(y) is 90 % of σ(y0), then the
uncertainty reduction is 10%; i.e. we have increased our knowledge on y by
10.

To reduce uncertainty for a target quantity, the observations need to reduce
uncertainty in the sub-space of the control space that (through the matrix
N′) projects onto the target quantity. In other words, it does not help to
constrain parts of the control space that have no impact on the target
quantity (as quantified by N′). Specific examples and further discussion of
how to interpret the matrix N′ will be provided in
Sect. below.

We note that (through Eqs. and ) the posterior
target uncertainty solely depends on the prior and data uncertainties as well
as the linearised model responses of simulated observation counterparts and
of target quantities. The approach does not require real observations and
can thus be employed to evaluate hypothetical candidate networks. Candidate
networks are defined by a set of observations characterised by observational
data type, location, time, and data uncertainty. Hence, the QND approach does
not require running the assimilation system. Here, we define a network as the
complete set of observations, d, used to constrain the model. The term
network is not meant to imply that the observations are of the same type or
that their sampling is coordinated. For example, a network can combine in
situ and satellite observations.

In practice, for pre-defined target quantities and observations, model
responses can be pre-computed and stored. A network composed of these
pre-defined observations can then be evaluated in terms of the pre-defined
target quantities without further model evaluation. Only matrix algebra is
required to combine the pre-computed sensitivities with the data
uncertainties. This aspect is exploited in our AOND system. The linearised
response functions were computed by the tangent linear version of NAOSIM
generated from the model's source code through the automatic differentiation
tool Transformation of Algorithms in Fortran
TAF;.

Experimental setupTarget quantities

The goal of this study is to explore the utility of the AOND system in
guiding observations for short-term to seasonal-scale sea ice predictions.
Ice forecasting at these timescales has been identified as a high priority
in the context of safe maritime operations
, management of marine living
resources and food security for indigenous communities
. Here, we focus on the first two issues in the Chukchi
and Beaufort seas north of Alaska (Figs. and
), which are experiencing some of the highest reductions in
summer ice concentration anywhere in the Arctic, along with major offshore
hydrocarbon exploration and potential impacts on protected species such as
walrus . Thus, the selection of target quantities for
the AOND system seeks to evaluate and improve predictions aimed at the
information needs of stakeholders and resource managers for this region. Of
particular interest is the summer season with its reduced ice cover. From an
observational point of view this period is particularly challenging, as
surface melt and its impact on ice dielectric properties complicate
retrievals of variables such as snow depth and ice thickness through
satellite remote sensing. For this study we deliberately selected the year
2007, a year of particularly low ice extent, which may be regarded as
representative of future ice conditions in a rapidly changing Arctic. As is
detailed in the following, we study both predictions for selected days and
predictions for integrals over selected time periods.

For all target regions delineated in Fig. , we use
spatial averages of the three simulated quantities: ice concentration
(fraction of area covered by ice, regardless of the 15 % floor used
in the definition of ice extent), ice thickness, and snow thickness. For each
of the target regions we look at these quantities for different days or time
periods. For the target region Chukchi Sea we examine these three quantities
for each of 10 April, 30 June, and 31 August, yielding a total of nine target
quantities. In order to specifically address information needs with respect
to safe shipping between Bering Strait and the central and eastern Beaufort
Sea (including supply of coastal communities and the oil industry hub at
Prudhoe Bay, offshore resource exploration and transits through the Northwest
Passage), we evaluate an additional set of target quantities derived from the
Barnett ice severity index (BSI). The BSI has developed into a standard
measure of shipping conditions and potential hazards encountered along the
Alaskan coast and at a critical choke point of the Northwest Passage, and it is
routinely issued by ice services . has
examined the predictive skill of statistical models in BSI seasonal
forecasts. The BSI is a composite of eight aspects of summer ice conditions
(see Table ), four related to the distance of the ice pack north
of Point Barrow (NOB) in mid-August and mid-September and four related to the
timing of ice retreat along the sea route from Bering Strait to Prudhoe Bay
during the entire navigation season (BS2PB). In replicating these variables
in a condensed way, we identify the two target regions as shown in
Fig. . The target region NOB covers a corridor of
50 km (one grid cell) width extending from Point Barrow to
75∘ N on 10 and 31 August. We use 31 August in contrast to
15 September (which is used in the definition of the BSI) because from the
end of August to mid-September 2007 the ice edge was located northwards of
75∘ N. For the region BS2PB, in keeping with the BSI we use the time
period from May to August.

Aspects entering the definition of the Barnett ice severity
index.

Distance from Point Barrow northward to ice edge (10 Aug).Distance from Point Barrow northward to ice edge (15 Sep).Distance from Point Barrow northward to boundary of 5/10 ice concentration (10 Aug).Distance from Point Barrow northward to boundary of 5/10 ice concentration (15 Sep).Initial date entire sea route to Prudhoe Bay less than/equal to 5/10 ice concentration.Date that combined ice concentration and thickness dictate end of prudent navigation.Number of days entire sea route to Prudhoe Bay ice-free.Number of days entire sea route to Prudhoe Bay less than/equal to 5/10 ice concentration.

In our variational assimilation system the largest possible control vector is
the superset of initial and surface boundary conditions as well as all
parameters in the process formulations. To keep our AOND system numerically
efficient, two- and three-dimensional fields are grouped into regions. We
proceeded by dividing the Arctic domain into nine regions
(Fig. ). In each of these regions we add a scalar
perturbation to each of the forcing fields (indicated in Table
by the type boundary “f”). Likewise we add a scalar perturbation
to five initial fields (indicated in Table by the type
initial “i”). For the ocean temperature and salinity the size of
the perturbation is reduced with increasing depth. Finally we have selected
18 process parameters from the sea ice–ocean model. This procedure resulted
in a total of 126 control variables, a superset of the set of control
variables identified by to have the largest impact on the
simulation. Unlike the study by the control vector used
here also includes process parameters. We conducted sensitivity experiments
in which we remove components from the control vector. For example, removing
the atmospheric forcing explores the (hypothetical) case of a perfect
seasonal atmospheric forecast, and removing the process parameters the
(hypothetical) case of a perfectly calibrated model.

The prior uncertainty of the control variables, C(x0) (see
Eqs. and ), is assumed to have diagonal form;
i.e. there are no correlations among the prior uncertainty relating to
different components of the control vector. The diagonal entries are the
square of the prior uncertainty (quantified by its standard deviation, in the
following denoted as SD or prior sigma). For process parameters this SD is
estimated from the range of values typically used within the modelling
community. The SD for the components of the initial state is based on a model
simulation over the past 20 years and computed for the 20-member
ensemble corresponding to all states on the same day of the year. Likewise
the SD for the surface boundary conditions is computed for the 20-member
ensemble corresponding to all 5-month forecast periods starting on the
same day of the year.

As the QND approach does not require the minimisation of Eq. (1), the
prior uncertainty only serves as a reference such that the impact of
observations is quantified in terms of a percentage change relative to the
prior uncertainty (uncertainty reduction). If the prior uncertainty were too
optimistic, the impact of the observations would be underestimated, and vice
versa: if the prior uncertainty were too high, the impact of the observations
would be overestimated. As we will use the same prior uncertainty as the
reference for all observational configurations, their relative performance is
not affected.

Observational networks

There are various types of observations sampling the Arctic sea ice–ocean
system, many of which are potentially suitable for assimilation into a model
like NAOSIMDAS. Our AOND system focuses on observations of ice concentration
(not used in the present study), snow depth and ice thickness. It provides
response functions for potential observations of each of these three
observables, for each surface grid cell, and for each day of the simulation
period (i.e. about 5 million possible observations of which subsets can be
selected for evaluation) with a user-defined data uncertainty. In this study
we demonstrate the application and potential utility of the system in
evaluating the relative merits and quantitative contribution to improving sea
ice forecasts for two alternate ice thickness airborne survey profiles. This
example is based on the need for objective guidance on flight routing as part
of NASA's Operation IceBridge, an airborne laser altimeter and snow radar
campaign meant to provide information on the mass budget of the Arctic ice
pack . Recent work has demonstrated the utility
of such data, collected in spring for initialisation and constraints on
seasonal forecasts of summer ice extent . Based
on an evaluation of flown and hypothetical IceBridge transects, we evaluate
the impact of simulated measurements along two transects within AOND. The
first is a transect from Bering Strait to Fram Strait, which we denote by
Chukchi to Fram (C2F, Fig. , red), and the second
from the Beaufort Sea to Fram Strait, which we denote by Beaufort to Fram
(B2F, Fig. , yellow). Both flights are assumed to
take place on 5 April 2007. The “observations” consist of model output of
ice and snow thickness at each grid cell that intersects with the transect as
indicated in Fig. . The default case specifies
a data uncertainty of 30 cm for both quantities. Sea ice
concentration is not observed. To explore the sensitivity of the results with
respect to the data uncertainty, we also test a data uncertainty of
10 cm. While the former is at the lower end of what is expected for
IceBridge altimeter data , the latter corresponds to the
lower bounds of airborne electromagnetic induction measurements
.

Results and discussion

Figure shows the performance of each transect in
improving forecasts over the Chukchi target region. We define the uncertainty
reduction relative to the case without observational constraints, where the
prior uncertainty in the control vector (see Sect. ) is
propagated to the three target quantities. Overall we note a larger impact of
C2F on the short-term forecast (10 days), while for B2F the impact
increases for the mid-term forecast (3 months). For the mid-term forecast C2F
surpasses B2F with respect to the impact on predicted ice concentration and
snow thickness, while its impact is marginally smaller for ice thickness. For
the 10 day forecast C2F has a much larger impact on predicted ice and
snow thickness than on ice concentration. This is mostly a result of the
flights observing specifically the former two quantities, whereas the model
dynamics require some time to transfer any constraints on snow and ice
thickness into constraints on ice concentration. Moreover, ice concentration
in this region is also strongly dependent on factors other than snow and ice
thickness, in particular during spring and early summer, when the role of wind
forcing greatly exceeds that of the other two variables.

Mathematically, through N′ in Eq. (), each target quantity
defines a one-dimensional sub-space (target direction;
) of the space spanned by the control vector (control
space). All control vectors v perpendicular to the target direction yield
N′v=0. Similarly, through M′ in Eq. () each observation
defines a second one-dimensional sub-space of the control space, the observed
direction. The better the observed direction projects onto the target
direction, the more efficient is the observation in reducing the uncertainty
in the target quantity. According to Eq. () the uncertainty
reduction increases with the response of the observable to a change in the
control vector (M′) and decreases with the data uncertainty.
Figure provides a visualisation of the complete
matrix N′, which shows the response of the three target quantities to
a change in each of the control variables by 1 SD of the prior probability
density function (Table ). The position on the x axis corresponds
to the number of the control variable in the last column of
Table . This provides two pieces of information: first, it shows
the target direction; second, it shows the size of the impact of an
uncertainty reduction in the target direction. We note that the initial
conditions of ice and snow have highest impact for the short-term forecast.
For the mid-term forecast, atmospheric forcing and model parameters gain in
importance. For the interpretation of the wind stress components taux and tauy
recall that the model operates on a rotated coordinate system. Taking the
rotation into account, for regions 6, 7, and 8
Fig. shows the direction in which a change of tau
yields the largest increase in ice thickness. Adding
a 25∘ Ekman deflection, the change of ice motion is towards the
target region. For the long-term forecast (153 days), the impacts (not shown)
are generally small because there is little ice left in the target area. The
impact of the B2F transect on the 10-day forecast of ice concentration over
the (remote) Chukchi target region (Fig. b) is
remarkable. It is explained by the relatively high impact of the lead closing
parameter h0 in the formulation of freezing (control variable #89) on ice
concentration (Fig. ). Since h0 is a global
parameter, observations on both transects can help to reduce uncertainty in
this parameter.

Figure shows the performance of each transect for
improving forecasts for the target region covering the coastal ocean from
Bering Strait to Prudhoe Bay (BS2PB). They show similar performance because
this target quantity is temporally averaged from May to August. B2F is
superior for snow thickness, and C2F for ice thickness and area. This can be
explained by the sensitivity of these three target quantities
(Fig. ). Relative to ice thickness and area, snow
thickness has a larger sensitivity to the initial (ice and snow) conditions
(in particular over region 8) than to the surface forcing and the process
parameters. And the initial snow thickness over region 8 is, of course,
better observed by B2F (which crosses this region) than by C2F. As an
additional test case we evaluate the combination of the two transects, which
clearly shows their complementarity.

Uncertainty reduction for target area BS2PB for flight
transects C2F, B2F, and both.

Figure shows the response of the three target
quantities to a 1 prior sigma change in each of the control variables. The
impact of wind stress dominates. For both region 7 and 8,
Fig. shows the direction in which a change of tau
yields the largest increase in ice thickness. Adding
a 25∘ Ekman deflection (to the right) the change of ice motion
is towards the intersection of the respective region's coastline with the
target area BS2PB. The parameter pstar has a positive impact because it yields
more rigid ice. Parameter h0, which essentially determines the
distribution of newly formed ice in the vertical vs. the horizontal
dimension, has a negative impact: increasing h0 yields thicker newly
formed ice and consequently reduces the ice concentration.

Figure shows the performance of each transect for improving
forecasts over the NOB target region. The performance of B2F is much better
than that of C2F for both forecast times. This result appears
counter-intuitive, because C2F is much closer than B2F, but can be explained
through the influence of the westward circulation prevailing in the waters
off the Alaskan coast . For forecast times of 4–5
months, an upstream observation is associated with much more predictive skill
than an observation directly over the target area. In fact the same mechanism
explains the previously mentioned higher uncertainty reduction of B2F for the
long-term forecast in the Chukchi area. For the target area BS2PB none of the
transects dominate because the target period is an integral from forecast
months 2 to 5.

Uncertainty reduction for target areas NOB for flight
transect B2F with data uncertainty of 0.1 m(a),
the assumption of perfectly known atmospheric forcing (b),
the assumption of a perfectly calibrated model (c),
the assumption of perfectly known atmospheric forcing and of a perfectly
calibrated model (d).

Figure shows the response of the three target quantities
(on both 10 and 31 August) to a 1 prior sigma change in each of the control
variables. We note the highest impact for tauy in region 8 (positive impact
of southwest increase), leading to more ice in the target region (see
Fig. ). Furthermore there is relatively high impact of
other atmospheric forcing variables, as well as of some parameters (the albedo of melting ice, albm, and the ice
strength parameter, pstar) and the ice initial conditions.

There is generally little difference in the responses for the two forecast
periods. This is an indication of the robustness of our linearisation of the
coupled sea ice–ocean system and confirms an analysis of
, who found, for the same model, moderate differences between the linearisation
and finite size perturbations. A consequence of this robustness is that the
specific target days we chose only play the role of a typical day within a
longer time period.

Figure shows the sensitivity of the performance of the
(superior) B2F transect with respect to changes in various impact factors
(relative to the default settings used for Fig. ) for the NOB
target region. The reduction in data uncertainty from 0.3 to 0.1 m
for both ice and snow thickness yields a considerable improvement in
performance (panel a). The effect is particularly pronounced for ice area.
Reducing the prior uncertainty for the atmospheric forcing to zero mimics the
availability of a perfect seasonal atmospheric forecast. Under this
assumption, the performance of the B2F transect is strongly increased (panel
b). Likewise a reduction of the prior uncertainty for all process parameters
mimics a perfectly calibrated model. Its effect on the performance of the B2F
transect is relatively small (panel c). Interestingly, combining the
perfectly calibrated model and the perfect atmospheric forecast assumptions
doubles the uncertainty reductions compared to the perfect atmospheric
forecast assumptions alone. In this case all the observational constraints can
fully act to reduce uncertainty in the initial conditions.

Conclusions

We have presented the AOND system that
evaluates hypothetical observational networks of the coupled sea ice–ocean
system in terms of their constraint on target quantities of interest within
an assimilation system. We have applied the tool to evaluate the potential of
two flight transects to reduce uncertainties in ice forecasts over periods
from 10 days to 5 months for regions with high offshore resource
exploration (Chukchi Sea) or shipping activity (Northwest Passage). For our
analysis and case study we selected the year 2007, a year of particularly low
ice extent, which may be regarded as representative of future ice conditions
in a rapidly changing Arctic.

Since our quantitative results are specific to the conditions in this
particular year, we focus on overarching conclusions that can be drawn from
this case study. First, we note that the network performance depends on the
specific question asked, i.e. on the target quantity. Of equal importance in the
highly advective Arctic sea ice regime is the finding that the longer the
forecast time, the further upstream we have to sample, well outside of the
region of interest. This may result in significant interannual variability in
the area that needs to be targeted for measurements relative to the region of
interest. This finding also supports the broader notion of an adaptive
sampling grid that reflects a priori knowledge of the state and dynamics of
the ice cover at the end of the ice growth season. On another level, we
furthermore demonstrated in a quantitative way how the model dynamics
transfer the observational information from one set of variables (snow depth
and ice thickness) to another variable (ice concentration). In this context,
we note that in our case study the target quantities and framework for
assessing the QND were based on the specific objective of predicting summer
ice conditions or navigation along a heavily trafficked route in the Alaskan
Arctic at the seasonal scale. Future work will have to evaluate the degree of
overlap in uncertainty reduction for predictions on seasonal as compared to
interannual or multidecadal timescales.

When defining candidate networks to be evaluated, it is essential to take
logistic constraints into account. The selection of alternate flight routes
for the C2F and B2F transects inherently reflects logistic factors. However,
the QND approach lends itself to inclusion of quantitative constraints on
specific regional data acquisition patterns that may require further work to
evaluate. Similarly, an essential input to the tool is the data uncertainty,
which is the combination of uncertainties in the observations and in
modelling their counterparts (model uncertainty). Hence, the QND approach can
also help in evaluating methodological improvements or evaluate the
costs/benefits of advances in instrumental design that reduce measurement
errors. These findings make it clear that a QND tool needs to be operated by
a team consisting of observationalists and modellers in order to derive
maximum benefits.

We note that the aforementioned model uncertainty to be provided to the tool
does not necessarily need to refer to the specific model that is used. As
long as the response functions of our model are approximately correct, we can
use the present system to simulate the observational impact on an
assimilation system around a different model. For QND results to be valid
beyond the model at hand, one has to employ a well-validated model that
includes all relevant processes. For example the model should have adequate
sensitivity of regionally integrated ice properties with respect to the
initial ice thickness. For the model used here this sensitivity is similar
for resolutions from 1/2 to 1/12 of a degree. One would not expect a drastic
change of this sensitivity when moving to even finer (eddy-permitting)
scales, but this requires further investigation. Computationally, the current
126-dimension control space requires 127 model simulations (over 5
months each) for the approximation of the Jacobian matrices (M′ of Eq. 3
and N′ of Eq. 4) quantifying observational and target sensitivities. This
should be feasible even for high-resolution models.

The current AOND system has the flexibility to also evaluate the potential of
space missions or further in situ sampling strategies. There are a number of
obvious ways to refine the present system. It can be extended to cover
climate conditions over longer timescales and further into the future,
possibly also representative of the state of the Arctic under climate change
scenario for mid-century and beyond. Moreover, one could add oceanic
observations or further target quantities, or extend the control vector to gain
broader insights into observing system design in the coupled atmosphere–sea
ice–ocean system. Furthermore, rather than operating Arctic-wide, the same
concept can be applied on a smaller regional scale when the forecasting period
is short enough to ensure that the main influence factors can be
appropriately simulated within the model domain.

Acknowledgements

This work has been funded by the European Commission through its Seventh
Framework Programme Research and Technological Development under contract
number 265863 (ACCESS) through a grant to FastOpt and OASys and by the
National Science Foundation as part of the Sea Ice Prediction Network (SIPN)
under grant number PLR-1304315. The authors thank two anonymous reviewers and
Christian Haas for valuable comments on the manuscript.
Edited by: C. Haas